Understanding Linear Inequalities: Practice Questions & Solutions
Practice solving linear inequalities with step-by-step solutions provided. Graph your solution set on a number line for better visualization. Improve your algebra skills with these practice questions.
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Presentation Transcript
CHAPTER 03 Algebra II Linear Relations Solutions: Practice Questions 3.4
03 Practice Questions 3.4 1. Solve the following inequalities and graph your solution set on a number line. x + 3 < 7, x (i) x + 3 7 (Subtract 3 from both sides) x 4 x
03 Practice Questions 3.4 1. Solve the following inequalities and graph your solution set on a number line. 5a 6 > 4, a (ii) 5a 6 4 (Add 6 to both sides) 5a 10 (Divide both sides by 5) a 2 a
03 Practice Questions 3.4 1. Solve the following inequalities and graph your solution set on a number line. 5 3p < 8, p (iii) 5 3p 8 (Subtract 5 from both sides) 3p 3 (Divide both sides by 3 and change the direction of the inequality) p> 1 p
03 Practice Questions 3.4 1. Solve the following inequalities and graph your solution set on a number line. 2x 4 > 10x, x (iv) 2x 4 10x (Subtract 2x from both sides) 4 8x (Divide both sides by 8) -4 8 > x 1 2 x x
03 Practice Questions 3.4 1. Solve the following inequalities and graph your solution set on a number line. 3z + 2 > z + 4, z (v) 3z + 2 z + 4 (Subtract z from both sides) 2z + 2 4 (Subtract 2 from both sides) 2z 2 (Divide both sides by 2) z 1 z
03 Practice Questions 3.4 1. Solve the following inequalities and graph your solution set on a number line. 7 3x 3 x, x (vi) 7 3x 3 x (Add 3x to both sides) 7 3 + 2x (Subtract 3 from both sides) 4 2x (Divide both sides by 2) 2 x x
03 Practice Questions 3.4 1. Solve the following inequalities and graph your solution set on a number line. (vii) x + 4 3(x 1), x x + 4 3(x 1) x + 4 3x 3 (Add 3 to both sides) x + 7 3x (Subtract x from both sides) 7 2x (Divide both sides by 2) 7 2 x x
03 Practice Questions 3.4 1. Solve the following inequalities and graph your solution set on a number line. (viii) 2(p + 4) < 2 p, p 2(p + 4) 2 p 2p + 8 2 p (Add p to both sides) 3p + 8 2 (Subtract 8 from both sides) 3p 6 (Divide both sides by 3) p<- 6 3 p 2 p
03 Practice Questions 3.4 1. Solve the following inequalities and graph your solution set on a number line. 3(k 3) < 2(k 1), k (ix) 3(k 3) 2(k 1) 3k 9 2k 2 (Add 9 to both sides) 3k 2k + 7 (Subtract 2k from both sides) k 7 k
03 Practice Questions 3.4 1. Solve the following inequalities and graph your solution set on a number line. 2(z 1) 5(z + 2), z (x) 2(z 1) 5(z + 2) 2z 2 5z + 10 (Subtract 10 from both sides) 2z 12 5z (Subtract 2z from both sides) 12 3z (Divide both sides by 3) 4 z z
03 Practice Questions 3.4 1. Solve the following inequalities and graph your solution set on a number line. 2x-5<1 3(x+3) (xi) 2x-5<1 3(x+3) (Multiply both sides by 3) 3(2x 5) < x + 3 6x 15 < x + 3 (Add 15 to both sides) 6x < x + 18 (Subtract x from both sides) 5x < 18 (Divide both sides by 5) 18 5 x < x < 3 6 x
03 Practice Questions 3.4 1. Solve the following inequalities and graph your solution set on a number line. x 2(x 1) + 5x > 3x 10 (xii) 2x 2 + 5x > 3x 10 7x 2 > 3x 10 (Subtract 3x from both sides) 4x 2 > 10 (Add 2 to both sides) 4x > 8 (Divide both sides by 4) x 2 x
03 Practice Questions 3.4 2. Write an inequality to represent each of the following shaded sections of the number line. Note that there may be more than one correct inequality to describe each solution set. (i) x 1, x or x > 2, x
03 Practice Questions 3.4 2. Write an inequality to represent each of the following shaded sections of the number line. Note that there may be more than one correct inequality to describe each solution set. (ii) x 1, x
03 Practice Questions 3.4 2. Write an inequality to represent each of the following shaded sections of the number line. Note that there may be more than one correct inequality to describe each solution set. (iii) 1 x 5, x x or x < 6, 1 x 5, x x or 0 < x < 6,
03 Practice Questions 3.4 2. Write an inequality to represent each of the following shaded sections of the number line. Note that there may be more than one correct inequality to describe each solution set. (iv) x x < 4,
03 Practice Questions 3.4 2. Write an inequality to represent each of the following shaded sections of the number line. Note that there may be more than one correct inequality to describe each solution set. (v) 3 x 1, x or 4 < x < 2, x
03 Practice Questions 3.4 2. Write an inequality to represent each of the following shaded sections of the number line. Note that there may be more than one correct inequality to describe each solution set. (vi) 3 < x 2, x
03 Practice Questions 3.4 3. Write down the values of ?that satisfy each of the following: x 2 < 6, where x is a positive, even number. (i) x 2 < 6 (Add 2 to both sides) x < 8 Since x is a positive, even number then x = 6, 4, 2
03 Practice Questions 3.4 3. Write down the values of ?that satisfy each of the following: x + 4 > 7, where x is a square number, less than 100. (ii) x + 4 > 7 (Subtract 4 from both sides) x > 3 Since x is a square number, less than 100 then x = 4, 9, 16, 25, 36, 49, 64, 81
03 Practice Questions 3.4 3. Write down the values of ?that satisfy each of the following: 2x 13 < 37, where x is a prime number. (iii) 2x 13 < 37 (Add 13 to both sides) 2x < 50 (Divide both sides by 2) x < 25 Since x is a prime number then x = 2, 3, 5, 7, 11, 13, 17, 19, 23
03 Practice Questions 3.4 3. Write down the values of ?that satisfy each of the following: 3x + 5 < 26, where x is a positive, odd number. (iv) 3x + 5 < 26 (Subtract 5 from both sides) 3x < 21 (Divide both sides by 3) x < 7 Since x is a positive, odd number then x = 5, 3, 1
03 Practice Questions 3.4 The width of a rectangle is 2x cm and its length is (10 x) cm, where x . 4. If the perimeter of the rectangle must be greater than 21 cm, find the smallest possible value of x. (i) 2(2x) + 2(10 x) > 21 4x + 20 2x > 21 2x + 20 > 21 (Subtract 20 from both sides) 2x > 1 x >1 2 Since x is a natural number, the smallest value x can be is 1.
03 Practice Questions 3.4 The width of a rectangle is 2x cm and its length is (10 x) cm, where x . 4. Hence, find the area of the rectangle for this value of x. (ii) Area = length width = (10 x) 2x (let x = 1) = (10 1) 2(1) = 9 2 = 18 cm2
03 Practice Questions 3.4 5. (i) Find the solution set of P: 2x 3 < 5, x (a) P: 2x 3 < 5 (Add 3 to both sides) 2x < 8 (Divide both sides by 2) x x < 4
03 Practice Questions 3.4 5. (i) Find the solution set of Q: 5x + 2 2x 1, x (b) Q: 5x + 2 2x 1 (Subtract 2x from both sides) 3x + 2 1 (Subtract 2 from both sides) 3x 3 (Divide both sides by 3) x 1 x
03 Practice Questions 3.4 (ii) Find P Q and graph your solution on the number line. 5. P Q: 1 x < 4 x x = 1, 0, 1, 2, 3
03 Practice Questions 3.4 Find the solution set M of 4 x < 6, x . 6. (i) M: 4 x < 6 (Add x to both sides) 4 < 6 + x (Subtract 6 from both sides) 2 < x, x
03 Practice Questions 3.4 (ii) Find the solution set Nof 3x 1 x + 9, x . 6. N: 3x 1 x + 9 (Add 1 to both sides) 3x x + 10 (Subtract x from both sides) 2x 10 (Divide both sides by 2) x 5, x
03 Practice Questions 3.4 6. (iii) If M N = a < x b, write down the value of a and the value of b. M N : 2 < x 5, x a = 2, b = 5
03 Practice Questions 3.4 6. (iv) Hence, find the value of a(a + b). a(a + b) = 2( 2 + 5) = 2(3) a(a + b) = 6
03 Practice Questions 3.4 Find the solution set P of 3(? 1) ?+ 8, ? . 7. (i) P: 3(x 1) x + 8 3x 3 x + 8 (Add 3 to both sides) 3x x + 11 (Subtract x from both sides) 2x 11 (Divide both sides by 2) x 5 5, x
03 Practice Questions 3.4 (ii) Find the solution set Qof 5(x + 2) > 2(x 1), x . 7. Q: 5(x + 2) > 2(x 1) 5x + 10 > 2x 2 (Subtract 10 from both sides) 5x > 2x 12 (Subtract 2x from both sides) 3x > 12 (Divide both sides by 3) x > 4, x
03 Practice Questions 3.4 (iii) If P Q = a < x b, write down the value ofa and the value ofb. 7. P Q: 4 < x 5 5, x a= 4,b = 5 5
03 Practice Questions 3.4 (iv) Hence, find the value of correct to one decimal place. (a2+b2) 7. a=-4, b=5 5 a2+b2= (-4)2+(5 5)2 = 16+30 25 = 46 25 =6 800735254 =6 8
03 Practice Questions 3.4 8. Given f (x) = 7x + 2 and g(x) = 3x + 22. Write down the values of x, for which f (x) < g(x), x . (i) f (x) < g(x) 7x + 2 < 3x + 22 (Subtract 2 from both sides) 7x < 3x + 20 (Subtract 3x from both sides) 4x < 20 (Divide both sides by 4) x x < 5 x = 1, 2, 3, 4.
03 Practice Questions 3.4 8. Given f (x) = 7x + 2 and g(x) = 3x + 22. Solve the inequality f (x) g(x) 6, x and, hence, graph the solution set on the number line. (ii) f (x) g(x) 6 (7x + 2) (3x + 22) 6 7x + 2 3x 22 6 4x 20 6 (Add 20 to both sides) 4x 26 x 6 5 x
03 Practice Questions 3.4 8. Given f (x) = 7x + 2 and g(x) = 3x + 22. Solve the inequality f (x) + 2g(x) 7, x and, hence, graph the solution set on the number line. (ii) f (x) + 2g(x) 7 (7x + 2) + 2(3x + 22) 7 7x + 2 + 6x + 44 7 13x + 46 7 (Subtract 46 from both sides) 13? 39 (Divide both sides by 13) x 3, x
